Stability of an integral curve of a second-order ordinary differential equation at the intersection of its singular set with the axis \(y = 0\)
Consider a second-order ordinary differential equation in the form of the system \[ a(x,y)\dot y=1,\dot x=y,a(x,y)\in\Omega^{1,n}, t\in T\subseteq \mathbb{R}. \] Its singular set of the first type is \(S_1=\{(x,y)\in \mathbb{R}^2:a(x,y)=0\}\) and \((x_0,0)=Y\cap S_1\) if \(x_0\) is the root of the equation \(a(x,0)=0\). Assume that the roots form a set of isolated points of the \(y\)-axis. In previous studies, the authors gave examples of large instability in the behavior of integral curves of second-order ordinary differential equations at the intersection of the singular set \(S_1\) with the axis \(y=0\). In earlier papers [see, for example: \textit{I. P. Pavlotsky} and \textit{M. Strianese}, Mat. Zametki, 75, 384--391 (2004; Zbl 1062.34001)], stability was analyzed only at singular points lying in either the half-plane \(y>0\) or the halt-plane \(y<0\). In this paper, a general stability condition is given for the points of \(S_1\) satisfying \(y=0\).
scientific article; zbMATH DE number 5357822